What are the roots of the polynomial equation g(x) = x^2 + 3x + 4x^2 + 4x + 29?

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Finding the Roots of the Polynomial

To find the roots of the polynomial equation g(x) = x^2 + 3x + 4x^2 + 4x + 29, we first rewrite the equation in standard form.

1. **Combine Like Terms**: The first step is to combine the like terms in the equation:

  • Combining the terms gives us:

    x^2 + 4x^2 + 3x + 4x + 29 = (1 + 4)x^2 + (3 + 4)x + 29
    = 5x^2 + 7x + 29

So, we rewrite our polynomial as g(x) = 5x^2 + 7x + 29.

2. **Using the Quadratic Formula**: To find the roots, we can utilize the quadratic formula:
x = (-b ± sqrt(b² – 4ac)) / (2a), where a = 5, b = 7, and c = 29.

3. **Calculate Discriminant**:
First, calculate the discriminant (D):

  • D = b² – 4ac = 7² – 4(5)(29) = 49 – 580 = -531

The discriminant is negative (D < 0), which indicates that there are no real roots.

4. **Finding Complex Roots**: Since the discriminant is negative, we can find complex roots:

  • x = (-7 ± sqrt(-531)) / (10)
  • We can express as i * sqrt{531}, where i is the imaginary unit.
  • The roots can then be expressed as:

x = (-7 ± i * sqrt{531}) / 10

5. **Conclusion**: The two complex roots of the polynomial equation g(x) = 5x^2 + 7x + 29 are:

  • x = (-7 + i * sqrt{531}) / 10
  • x = (-7 – i * sqrt{531}) / 10

These roots indicate that the polynomial does not cross the x-axis, confirming the absence of real roots and the presence of complex roots instead.

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